Primary Kodaira surfaces

A primary Kodaira surface X = (M, I) is a compact complex surface with κ(X) = 0, b₁(X) = 3, c₁(X) = 0, c₂(X) = 0. Moreover the other numerical characters of X are given as follows:

h^1,0(X) = 1, q(X) = 2, p_g(X) = 1, b⁺₂(X) =b⁻₂(X) = 2,

where h^1,0(X), q(X) and p_g(X) denote the complex dimension of the space of holomorphic one-forms, the irregularity and the geometric genus of X, respectively (see Barth et al. [6]). Any primary Kodaira surface admits no positive-definite K¨ahler metric, since its first Betti number b₁(X) is three.

It is well-known that every primary Kodaira surface X is covered by the complex planeC² and its fundamental groupπ₁(X) is represented injectively into the complex affine transformation group Affine(C²) on C²:

ρ:π₁(X)−→Affine(C²), γ →ρ_γ, ρ_γ(z₁, z₂) = (z₁+α_γ, z₂ +α_γz₁+β_γ),

where (z₁, z₂) are the standard complex coordinates of C² and α_γ, β_γ are constants in C depending only on γ. Setting G := ρ(π₁(X)), we can then identify X with C²/G, as a complex surface (see Kodaira [51]).

We are now in a position to state one of our main results in this chapter.

Theorem 3.16 Let X = C²/G be a primary Kodaira surface. Then the following two-forms Ω₁,Ω₂,Ω₃ define a neutral hyperk¨ahler structure on X:

Ω₁ = Im(dw₁∧dw₂) +√

−1Re(w₁)dw₁ ∧dw₁+ (√

−1/2)∂∂ϕ, Ω₂ = Re(e^√−1θdw₁∧dw₂), Ω₃ = Im(e^√−1θdw₁∧dw₂), (3.15)

where (w₁, w₂) is the standard complex coordinate system of C², θ is a real constant and ϕ is a solution of the equation

4√

−1(Im(dw₁∧dw₂) +√

−1Re(w₁)dw₁∧dw₁)∧∂∂ϕ=∂∂ϕ∧∂∂ϕ.

(3.16)

In particular, any primary Kodaira surface admits a neutral hyperk¨ahler structure. Conversely, under suitable complex coordinates (w₁, w₂) of C², the fundamental form of any neutral hyperk¨ahler structure onX is expressed as (3.15).

Proof. Let Ψ : X → ∆ be an elliptic fiber bundle structure over the base elliptic curve ∆. Then we have the following commutative diagram:

C² −−−−→ X

Ψ



 ^Ψ C −−−−→

∆

where Ψ is the projection from C² to the first factor C, and , are the covering maps. Let (z₁, z₂) denote the standard complex coordinate system of C² above. Then φ:=dz1 gives rise to a nonvanishing holomorphic one-form on X, and generates the cohomology group H⁰(X; Ω¹_X) ∼= H^1,0

∂ (X), where Ω¹_X denotes the sheaf of germs of holomorphic one-forms onX. Furthermore, σ^0,1 := dz₂ −z₁dz₁ is a ∂-closed (0,1)-form on X, and the ∂-cohomology classes of ¯φand σ^0,1 generate the Dolbeault cohomology groupH¹(X;OX)∼= H^0,1

∂ (X), where OX denotes the structure sheaf of X. Since dσ^0,1 =−dz₁∧ dz₁, a real one-form σ := σ^0,1 +σ^0,1 on X is d-closed. Furthermore, we see that the cohomology classes of φ,φ¯ and σ generate H¹(X;C). Note that dz1∧(dz2−z1dz1) yields a nonvanishing holomorphic two-form on X.

Let (Ω₁,Ω₂,Ω₃) := (Ω_I,ΩJ,ΩK) be the fundamental form of a neutral hyperk¨ahler structure (g, I,J, K) on X =C²/G. As mentioned in Proposition 3.15, Ω₂+√

−1Ω₃ is a nonvanishing holomorphic two-form onX, and hence defines a global section of the canonical bundle K_X. Therefore there exists a nonzero constant c₀ =|c₀|e^√−1ψ ∈C (ψ ∈R) such that

Ω2 +√

−1Ω3 =c0dz1∧(dz2−z1dz1), since X is compact.

Now, define reald-closed two-forms Ω⁻₂ and Ω⁻₃ respectively by Ω−2 +√

−1Ω−3 :=√

−1dz1∧(dz2 −z1dz1).

It follows from Proposition 3.6 and the definitions of Ω⁻₂ and Ω⁻₃ that

−Ω²₁ = Ω²₂ = Ω²₃ =−|c₀|²(Ω⁻₂)² =−|c₀|²(Ω⁻₃)², Ω−2 ∧Ω−3 ≡0, Ωa∧Ω−b ≡0 (2≤a, b≤3).

(3.17)

We then verify that (|c₀|Ω⁻₂,Ω₂,Ω₃) and (|c₀|Ω⁻₃,Ω₂,Ω₃) define neutral hy- perk¨ahler structures onX, respectively. Note that the cohomology classes of Ω2,Ω3,Ω−2,Ω−3 generate the cohomology groupH²(X;R) and satisfy relations similar to those in (3.17). Recall that the K¨ahler form Ω₁ is a closed real (1,1)-form on X and its cohomology class [Ω₁] in H²(X;R) is orthogonal to [Ω2] and [Ω3] with respect to the cup product. Hence there exist a real one-form η and real constants a, b such that

Ω₁ =|c₀|(aΩ⁻₂ +bΩ⁻₃) +dη.

It then follows from (3.17) that

(1−a²−b²)Ω₁² =d(η∧[2|c₀|(aΩ⁻₂ +bΩ⁻₃) +dη]).

By integrating the equation above, we obtain a² +b² = 1, so we may set a = cos and b= sin for some real constant .

Recalling the decompositionη =η^1,0+η^0,1 (η^0,1 =η^1,0), we see that η^0,1 is ∂-closed, since Ω1,Ω−2,Ω−3 are real (1,1)-forms, and hence that

η^0,1 =kφ+lσ^0,1+∂µ, dη= (¯l−l)dz₁∧dz₁+∂∂(µ−µ),¯

wherekandlare constants, andµis a complex-valued function onX. Setting

√−1c|c0|^2/3 := ¯l−l (c∈R) and √

−1ϕ := 2(µ−µ), we then see that¯ Ω₁ =|c₀|(cosΩ⁻₂ + sinΩ⁻₃) +√

−1c|c₀|^2/3 dz₁∧dz₁+ (√

−1/2)∂∂ϕ.

By making use of the coordinates

(w₁, w₂) := (|c₀|^1/3e^√−1z₁+c,|c₀|^2/3z₂), we can express Ω₁,Ω₂,Ω₃ as

Ω1 = Ω0+ (√

−1/2)∂∂ϕ, Ω₂ +√

−1Ω₃ =e^{√−1(ψ−)}dw₁∧dw₂ =:e^√−1θdw₁∧dw₂, where Ω₀ is given by

Ω₀ := (√

−1/2)(dw₁∧dw₂−dw₁∧dw₂+ (w₁ +w₁)dw₁∧dw₁).

Therefore we see that (Ω₁,Ω₂,Ω₃) defines a neutral hyperk¨ahler structure on X if and only if ϕ satisfies the following equation:

4√

−1Ω₀∧∂∂ϕ=∂∂ϕ∧∂∂ϕ.

This completes the proof.

We note that the corresponding metric g =g_ϕ is explicitly given by g_ϕ = (w₁+w₁)|dw₁|²−(dw₁dw₂ +dw₁dw₂) +D²ϕ,

(3.18)

where D²ϕ denotes the complex Hessian of ϕ. Clearly, the pull-back of any function on the base torus ∆ is a solution of (3.16).

By using the expression (3.15), we may give a characterization of flat neutral hyperk¨ahler structures on a primary Kodaira surface in terms of the potential function ϕ, which shows that each nonconstant function ϕ on the base torus of any primary Kodaira surface defines a non-flat neutral hyperk¨ahler metric gϕ (cf. Petean [82]).

Theorem 3.17 Let g_ϕ be the neutral hyperk¨ahler metric on a primary Ko- daira surface X defined by (3.18), where ϕ is a solution of (3.16). Then g_ϕ is flat if and only if ϕ is constant.

Proof. LetX =C²/Gbe a primary Kodaira surface,g a neutral hyperk¨ahler metric on X, and (Ω₁,Ω₂,Ω₃) the fundamental form. In terms of complex coordinates (w₁, w₂) satisfying Ω₂ +√

−1Ω₃ = e^√−1θdw₁ ∧dw₂ (θ is a real constant), the condition −Ω²₁ = Ω²₂ = Ω²₃ is written as

g_1¯1g_2¯2−g_1¯2g_2¯1 ≡ −1.

(3.19)

Thus the components g^αβ¯ satisfy

g^¯11 =−g_2¯2, g^¯12=g_1¯2, g^¯21 =g_2¯1, g^¯22 =−g_1¯1. The connection form {ω^α_β} is given by

ω₁¹ =−g_2¯2∂g_1¯1+g_2¯1∂g_1¯2, ω₂¹ =−g_2¯2∂g_2¯1+g_2¯1∂g_2¯2, ω₁² = g_1¯2∂g_1¯1−g_1¯1∂g_1¯2, ω₂² = g_1¯2∂g_2¯1−g_1¯1∂g_2¯2. (3.20)

In particular, it follows from (3.19) that ω₁¹+ω²₂ ≡0.

(3.21)

Recall that the fundamental form Ω₁ may be written as Ω₁ = (√

−1/2)(−dw₁∧dw₂−dw₂ ∧dw₁+ (w₁+w₁)dw₁∧dw₁+∂∂ϕ), where ϕ is a certain smooth function on X. The components g_αβ¯ are given explicitly by

g_1¯1 =w₁+w₁+ ∂²ϕ

∂w₁∂w₁, g_1¯2 =−1 + ∂²ϕ

∂w₁∂w₂(= g_2¯1), g_2¯2 = ∂²ϕ

∂w₂∂w₂. From (3.20) and (2.17), we see that g is flat if ϕ is constant.

For anyγ ∈G, we define ρ_γ :C² −→C² by

ρ_γ(w₁, w₂) = (w₁+α_γ, w₂+α_γw₁+β_γ).

It then follows that

ρ^∗_γ(dw₁) =dw₁, ρ^∗_γ(dw₂) =dw₂+α_γdw₁, (3.22)

ρ_γ∗(∂₁) = ∂₁+α_γ∂₂, ρ_γ∗(∂₂) = ∂₂. (3.23)

Then we can verify the following relations:

g_1¯1◦ρ_γ =g_1¯1−α_γg_1¯2−α_γg_2¯1+|α_γ|²g_2¯2, g_2¯2◦ρ_γ =g_2¯2, g_1¯2◦ρ_γ =g_1¯2−α_γg_2¯2, g_2¯1◦ρ_γ =g_2¯1−α_γg_2¯2. (3.24)

By making use of these relations, we also have

ρ^∗_γω₁¹ =ω₁¹−α_γω₂¹, ρ^∗_γω¹₂ =ω₂¹, ρ^∗_γω₁² =ω²₁+ 2α_γω₁¹−α_γ²ω¹₂. (3.25)

If we set

η₁ :=ω₁¹+w₁ω₂¹, η₂ :=ω¹₂, η₃ :=ω₁²−2w₁ω₁¹−w₁²ω¹₂, then η₁, η₂, η₃ may be regarded as one-forms on X =C²/G.

In what follows, we suppose that g is flat. Then η₂ is a holomorphic one-form on X. Since h^1,0(X) = 1, we can write η₂ as

η₂ =Adw₁, where A is a constant. In particular,

dη₂ =∂η₂ =∂η₂ ≡0.

Lemma 3.18 η2 ≡0.

Proof. From the flatness of g and (3.21), we have

0≡dη₂ =dω₂¹ =−(ω¹₁∧ω₂¹+ω₂¹∧ω₂²) =−2ω₁¹∧ω¹₂. Thus we also have

η1∧η2 = (ω¹₁ +w1ω¹₂)∧ω₂¹ ≡0.

IfA= 0, thenη₁∧dw₁ ≡0. Sinceη₁ is a (1,0)-form onX, we have a function F onX such that

η1 =F dw1, i.e., ω¹₁ = (F −Aw1)dw1. By the flatness of g again, we then obtain

0≡∂ω₁¹ = (∂F −Adw₁)∧dw₁.

Namely, we see that ∂F =Adw₁ and hence ∂∂F ≡0. From the mean value property for the operator ∂∂, we then conclude that F must be constant.

Thus Adw1 = ∂F ≡ 0, that is, A = 0. This contradicts the assumption

A= 0.

It follows from Lemma 3.18 and (3.25) that there exists a constant B such that

η₁ =Bdw₁. Then it is easy to see that

∂η₃ = 2Bdw₁∧dw₁, ∂η₃ = 2Bdw₁∧η₃. (3.26)

We may assume that η₃ is expressed as

η3 =f1dw1 +f2(dw2 −w1dw1)

for smooth functions f₁, f₂ on X. It then follows from (3.26) that

∂(f₁ −w₁f₂) + 2Bdw₁ ≡0, ∂f₂ ≡0.

(3.27)

In particular, f₂ is a holomorphic function on X, and must be a constant, say C. It follows from (3.27) that

∂∂f₁ =∂((−2B+C)dw₁)≡0.

From the mean value property for ∂∂ again, we see that f₁ is also constant, say K. It is then easy to see from (3.26) that

2Bdw₁∧dw₁ =∂η₃ =∂(Kdw₁+C(dw₂−w₁dw₁)) =Cdw₁∧dw₁, 2BCdw₁∧dw₂ =∂η₃ =∂(Kdw₁+C(dw₂−w₁dw₁))≡0, and hence B =C = 0. Thus we obtain

η₁ =η₂ ≡0, η₃ =Kdw₁. Using (3.19) and (3.20), we also have

∂g_2¯2=−∂g_2¯1 ≡0, ∂g_1¯2=−Kg_2¯2dw₁, ∂g_1¯1=−Kg_2¯1dw₁. In particular, g_2¯2 is a constant, since ∂g_2¯2 =∂g_2¯2 ≡0.

By integrating g_2¯2 = ∂²ϕ/∂w2∂w2 on each fiber T of Ψ : X −→ ∆, we obtain

g_2¯2

T

dw₂∧dw₂ =

T

∂²ϕ

∂w2∂w2dw₂∧dw₂ = 0,

and hence g_2¯2 ≡0. Thus ϕ depends only on the variable w₁, so that ϕ may be regarded as a function on ∆. In particular, g_1¯2 =g_2¯1≡ −1. On the other hand, we can regard g_1¯1−(w₁+w₁) as a function on X, satisfying

∂∂(g_1¯1−(w₁ +w₁)) =−∂(∂g_1¯1−dw₁) =−∂(K −1)dw₁ ≡0.

Hence g_1¯1−(w₁+w₁) must be constant, sayL. IntegratingL=∂²ϕ/∂w₁∂w₁ on ∆, we also haveL= 0. Thereforeϕ is constant. Namely,g must coincide

with g0.